The differential equation of all circles passing through the origin a…

The equation of the family of circles passing through the origin and having their centres on x-axis is

$(x-a)^2+(y-0)^2=a^2$ or, $x^2+y^2-2 a x=0$

Differentiating w.r. to x, we get

$2 x+2 y \frac{d y}{d x}-2 a=0 \Rightarrow a=x+y \frac{d y}{d x}$

Substituting the value of a in (i), we get

$x^2+y^2-2 x^2-2 x y \frac{d y}{d x}=0$ or, $y^2=x^2+2 x y \frac{d y}{d x}$

as the required differential equation.